Arithmetic symplectic geometry via mirror symmetry?

Question

Homological mirror symmetry in the classical setting relates the bounded derived category of coherent sheaves on a Calabi-Yau manifold to the split-closure of the derived Fukaya category of the mirror Calabi-Yau.

In the paper 'Arithmetic mirror symmetry for the 2-torus', authors construct a $\mathbb{Z}$-linear equivalence between exact Fukaya category of a punctured torus and the category of perfect complexes of coherent sheaves on the central fiber of Tate curve.

Intuitively, this statement is only a half of the HMS conjecture since we also would like to have an equivalence between the 'Fukaya category of the central fiber of Tate curve' and the bounded derived category of coherent sheaves on the punctured elliptic curve.

The question is: is it possible to somehow define the notion of Fukaya category for the central fiber of the Tate curve (which is a curve in $\mathbb{P}^2(\mathbb{Z})$, not a smooth manifold)? If it is possible, can we construct an equivalence to the derived bounded category of coherent sheaves on the punctured elliptic curve?

Answer 1

Yes. Recently Auroux (jointly with Efimov and Katzarkov) has proposed a definition of the Fukaya category for trivalent configurations of rational curves. If $\Sigma_g$ is a genus $g$ Riemann surface with $g\geq2$, then its mirror is a trivalent configuration of $3g-3$ rational curves meeting in $2g-2$ triple points. In the case when $g=1$, a nodal curve is obviously not a trivalent configuration. However, one can replace it with a nodal curve with an affine line attached at the node, which then enables one to make sense of its Fukaya category. The objects of this version of Fukaya category are embedded graphs with trivalent vertices at the triple points, and morphisms are linear combinations of intersection points as in usual Floer theory. Note that since the punctured elliptic curve $E^\circ:=E\setminus\{pt\}$ is not compact, there are two versions of derived categories of coherent sheaves, namely one can consider either the usual $D^b\mathit{Coh}(E^\circ)$ or its compactly supported version $D^b\mathit{Coh}_\mathit{cpt}(E^\circ)$. On the mirror side, this determines whether one needs to do wrapping on the affine line component or not.